The Explanatory Power of Functions of the Variables - Research Paper Example

Topic: 2 Table of Contents Introduction 3 Theoretical Model 4 Hypothesis Testing and Diagnostic Tests 5 Conclusion 10 Frank, H. & Althoen, S. C. (1994) Statistics: concepts and applications. USA: Cambridge University Press. 11 Gravetter, F. J. & Wallnou, L. B. (2004). Statistics for the Behavioral Sciences (8th Edition). Cambridge: Wadsworth. 11 Bibliography 11 Introduction The recent financial crash has afflicted a large number of economies globally due to its devastating consequences. Many a nation has been forced to reconsider their previous policy measures and reshuffle the composition of their expenses. The intensity of the effect is especially felt in the developing economies which have but a small stock of resources with which to revive their frail economic system. Potency of the crisis seems to be cumbersome particularly because of the financial system of an economy being the more affected segment of the economy. Financial sector happens to be the crux of an economic system, responsible for a robust functioning of various states of affairs. Since the proper functioning of an economy is dependent upon the efficient operation of the domestic financial sector, so, a crisis-affected financial system of an economy needs to be attended readily. A robust financial sector can bail a nation out of many crucial phases which makes its well-being of prime importance for the administration of the nation. On the other hand, education is another factor which has the capacity to brighten the future prospects of the nation, through educating the national youth and training them with special skills, thus enhancing the quality of human capital. Vocational training which forms a part of secondary education in any nation is an integral source for the creation of employment opportunities for the national youth and thus slashing the unemployment rate in the economy. Apart from vocational training, secondary education also guides the national youth towards a path of better employment opportunities. For a developing nation, both aspects, viz., attending to the shock-stricken financial sector as well as framing policies for the betterment of the domestic education sector, have almost equal significance. The only difference lies in the time span necessary to enjoy the outcomes of their efficient operation. Stabilising the domestic financial sector will have a short-term impact as it can save other segments of the nation from subsiding as well; whereas, augmenting the prospects of the education sector will help to fight back future worries on account of growing unemployment and recessionary trends in the nation. The national youth will be capable enough to take care of their respective futures. Since sponsoring each of the factors will bring significant contributions towards economic growth and development aspects, it is a rather tricky situation that the national authorities have in front of them, given that the resources allotted for investment is limited, viz., US$ 2 million. Objectives of the present paper will be to deduce the factor which can bring the maximum contribution towards economic development prospects and thus be the most beneficial for the government of any nation. The idea is to find out the explanatory power of functions of each of the two variables and choose the one which can influence the national growth structure in the most effective way. Theoretical Model The economic model in order to seek out the most appropriate avenue for investing the administration’s limited funds is, dlypci = 1 + 2lypc90i + 3lsecedi + 4govgdpi + 5openi + 6infli + 7crediti + ui Where, ypc05 = per capita GDP (constant prices, chain series) in 2005 ypc90 = per capita GDP (constant prices: chain series) in 1990 dlypc = ln(ypc05) – ln(ypc90) lypc90 = ln(ypc90) seced = percentage of secondary school age population enrolled at secondary school in 1990 lseced = ln(seced) govgdp = government share of real GDP per capita in 1990 open = openness = ratio of (exports + imports) to GDP in 1990 cpi90 = consumer price index value in 1990 cpi85 = consumer price index value in 1985 infl = five-year inflation rate = ln(cpi90) – ln(cpi85) credit = ratio of private credit by deposit money banks and other financial institutions to GDP in 1990 Hypothesis Testing and Diagnostic Tests The purpose of the regression model will be to analyse the exact channel through which growth aspects of the nation will be maximised. The factor which has the most significant explanatory power will be the most suitable investment avenue. In order to find out the whether the explanatory powers of the independent variables, found out by the estimated model is significant or not, it is necessary to conduct certain hypothesis testing, where the hypotheses are defined as follows – H01: 2 = 3 = 4 = 5 = 6 = 7 = 0 against H11: j  0 for at least one j  (2...7) H02: 02 = 0 against H12: 2  0 using a significance level of 0.05. H03: 03 = 0 against H13: 13 > 0 using a significance level of 0.05. H04: 07 = 0 against H14: 17 > 0 using a significance level of 0.1. The first one of the hypotheses aims to find out if the model being estimated is a significant one, i.e., whether the explanatory power of the predicted model is enough to explain variations in the dependent variable. If all the estimated coefficients are found to assume a value equal to zero, it indicates that none of the corresponding variables render any significant impact on the dependent variable, ‘dlypc’. The ideal test to examine this hypothesis will be an F-test, which is calculated on the basis of ANOVA results (Frank & Althoen, 1994). F-statistic is defined as, F = MSE/ MSR Where, MSE = Mean sum of squares of the estimated model =  Explained Sum of Squares/ degrees of freedom and MSR = Mean sum of squares of the estimated residuals = Residual sum of squares/ degrees of freedom. The remaining three hypotheses test for the significance of the explanatory variables of the corresponding variables. Quite evidently, if the estimated coefficients are found to be equal to zero, then the accompanying variables signify no importance in explaining the variations in the dependent variable. The ideal test to examine the aforementioned hypotheses is Student’s t-test, defined as, T = β* - β0 / se (β*) Where, β0 = Hypothesised population value of coefficient β, β* = Estimated value of coefficient β and se (β*) = Standard error of estimated coefficient β. The Student’s t-test is thus, an attempt to find out whether any significant difference exists between the estimated and the hypothesised population value of the parameter (Kurtz, 1963). After estimation, the statistics have to be compared with their tabulated values so as to reject or accept the respective null hypotheses - in other words, to find whether the estimated statistics are significant or not. Rules for the acceptance or rejection of null hypotheses are based on a comparison of the predicted statistic with their tabulated values at given degrees of freedom and standard levels of significance. A modified criterion, made easy due to the use of statistical software STATA, is through a comparison of the estimated levels of significance with the standard ones.1 1. If estimated level of significance of the parameter is greater than the standard level of significance, α, then the null hypothesis cannot be rejected at (α x 100)% level of significance or rather with a (1 – α) x 100% confidence. 2. If estimated level of significance of the parameter is lower than the standard level of significance, α, then the null hypothesis can be rejected at (α x 100)% level of significance or rather with a (1 – α) x 100% confidence. The above rules however are applicable only in cases of two-tailed tests. The change in cases of one-tailed tests is that, the estimated level of significance is halved before comparing. Estimation of the regression model does not complete the analytical task, since it still remains to be seen whether the model complies with the features of the Classical Linear Regression Model or not. If they are not found to go in line with the model characteristics, then the regression being fitted is considered to be more like a sampling bias, rather than a true model. Three basic assumptions of CLRM are – linearity, homoscedasticity and normality (Gravetter & Wallnou, 2004). Tests used to examine their validity are known as diagnostic tests, three of which have been discussed below. 1. Linearity feature assumes that the fitted model must be a linear one, i.e., it must be devoid of an outlier problem, which can ideally be detected through running a regression of the previous explanatory variables along with a non-linear function of the predicted dependent variable on the predicted residual. If the non-linear variable is found to be explaining variations in the predicted error term well, then assumption of linearity is considered to be violated. 2. Homoscedasticity assumption assumes that the estimated residuals are placed equally from their mean values and thus facilitates to think that the residuals are not influenced by the dependent variable. An ideal way of correcting it is through estimating a regression model of squares of the predicted residual terms on squares of the predicted dependent variable. If the coefficient of the latter is found to be a significant one, then homoscedasticity assumption is supposed to be violated. 3. Normality assumption of the estimated residuals helps to estimate the population parameters in a definite manner, which is where the importance of the assumption comes from. Skewness-Kurtosis test for normality heck is highly suitable for the purpose. A variable deviating from normality exhibits a skewness statistic significantly different from 0, while a kurtosis statistic different from 3. Results and Interpretation The regression model being estimated is found to be, dlypci = 0.7566 – 0.1998lypc90i + 0.2503lsecedi + 0.0100govgdpi + 0.0002openi + 0.1143infli + (0.068) (0.011) (0.013) (0.117) (0.922) (0.287) 0.2125crediti + ui (0.136) For null hypothesis, H02, estimated level of significance of 2 = 0.011 < 0.05, H03, estimated level of significance of 3 = 0.0065 < 0.052 H04, estimated level of significance of 7 = 0.068 < 0.12 Hence, null hypothesis is rejected at the given levels of significance, in each of the three cases, implying that each one of the corresponding variables have significant implications on the dependent variable. Similarly, for null hypothesis H01, estimated level of significance of F-statistic = 0.1289 > 0.05. Hence, the null hypothesis in this case cannot be rejected at 5% level of significance, implying that the variables being used in the model are not sufficient to explain variations in the dependent variable. Given the above model, it is found that, the Finance Ministry would be wise to invest in the secondary education sector rather than in the financial sector of the nation. Linearity Assumption Estimating the regression model of predicted residuals on all the previous independent variables along with square of the predicted dependent variable shows that the latter cannot explain variations in the dependent variable significantly, implying that the estimated residuals are linear in nature. This conclusion had been drawn after closely examining the p-value of the estimated coefficient of the specified variable. Homoscedasticity Assumption The estimated coefficient of squared predicted dependent variable when estimated on the predicted residual of the original equation, is found to be rather insignificant (clear from p-value statistic), implying that the assumption of homoscedasticity of estimated residuals is not violated. Normality Assumption The estimated residuals when exposed to a normality test, reveal a highly negative skewness and a kurtosis far greater than 3, implying that the residuals do not follow a normal distribution. Hence, it indicates towards the presence of outliers and implies that a dummy variable must be included and the regression must be re-estimated to get rid of the problem. Here, the dummy variable is defined as, D = 1, for the observation having an outlier and 0, otherwise. After incorporating a dummy and re-estimating the model, it is found to assume the following form, dlypci = 0.7140 – 0.1909lypc90i + 0.2600lsecedi + 0.0082govgdpi + 0.0003openi + 0.0907infli + (0.039) (0.004) (0.002) (0.121) (0.813) (0.309) 0.1564crediti – 1.062D+ ui (0.187) (0.000) Repeating the diagnostic tests, it is found that, all the three assumptions being considered, are satisfied (following p-value rule, as specified above), implying that the newly estimated model is the correct one.3. In fact, the effectiveness of investing in secondary education is higher than that in the financial sector, as is evident from the following pair of equations. Conclusion The empirical analysis conducted in the paper throws some light upon the possible government strategy to ensure economic development in the long-run. After re-estimation, it is quite clear that spending on the secondary sector can yield a better growth opportunity (0.027) than spending on the financial sector (0.022) can, and thus, the latter must be discarded as a potential avenue for the flow of resources. References Frank, H. & Althoen, S. C. (1994) Statistics: concepts and applications. USA: Cambridge University Press. Gravetter, F. J. & Wallnou, L. B. (2004). Statistics for the Behavioral Sciences (8th Edition). Cambridge: Wadsworth. Heston, A., Summers, R. and Aten, B. (August 2009) Penn World Table Version 6.3, Center for International Comparisons of Production, Income and Prices at the University of Pennsylvania. Kurtz, T. E. (1963). Basic statistics. USA: Prentice-Hall. Bibliography Levin, J. & Fox, J. A. (1993). Elementary Statistics in Social Research. USA: Pearson. Appendix 1. Regression Estimation (Original) Diagnostic Tests Linearity Assumption Homoscedasticity Assumption Normality Assumption 2. Regression after incorporating dummy Diagnostic Tests Linearity Assumption Homoscedasticity Assumption Normality Assumption Read More